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370      XV   NORMED ALGEBRAS AND SPECTRAL THEORY

Let Hk be the closure of the vector subspace of H generated by the vectors
V(s)-x where skeak and xeH. Since each ye A can be written in the
form s = £Sk with skeak, and since V is continuous, we have

(5.5.2), and therefore H is the closure of the sum of the Hk . Also, ifh^k and
^ea*, Jkeafc,wehave

(F(s,) • x | 7(5*) • y) = (V(s%Sh) -x\y)=0

because a, is self-adjoint and akah = {0}. This proves (i). Now assume that A
is topologically simple and finite-dimensional over C, and therefore has a
unit element; we can then restrict ourselves to the case where there exists a
totalizer x0 for V (1 5.5.6). The vector subspace of H generated by the V(s) - XQ
is finite-dimensional, hence closed (5.9.2) and so equal to H. We may there-
fore argue by induction on the dimension of H. Since A is the sum of a finite
number of minimal left ideals, there is at least one minimal left ideal, say I,
such that the subspace E = F(I) • x0 is nonzero. The surjection s^ V(s) - x0
of I onto E is then an A-module homomorphism, and since its kernel is a left
ideal I' contained in I and distinct from I, we have I' = {0}. Hence E is a
stable subspace of H with respect to K, such that the restriction of K to E is
equivalent to U{ . Since the orthogonal supplement H' of E in H is stable with
respect to V and of dimension strictly less than the dimension of H, we have
only to apply the inductive hypothesis to complete the proof.

The result of (1 5.8.1 6(ii)) can be shown to be valid without assuming that
afc is finite-dimensional (Problem 1).

PROBLEMS

1.   Let A be a topologically simple, separable, complete Hilbert algebra.

Let K be a nondegenerate representation of A in a Hilbert space H. With the
notation of the proof of (15.8.14), put En = V(en) and An = V(an). The En are orthogonal
projectors on subspaces Hn of H, such that H is the Hilbert sum of the Hw, and we
have ,4n(Hi) = Hn and ^?(Hn) = Hi. If (6*0*ei is a (finite or infinite) orthonormal
basis of HI, then for each n the vectors bkn =* y~il2An(bki) form an orthonormal basis
for Hn as k runs through the index set I. Deduce that, if H* is the subspace of H gener-
ated by the bkn (n ^ 1), then Hi is stable with respect to V, and V is the Hilbert sum
of the restrictions Vk of V to Hi. Each of those representations is topologically cyclic,
bki being a totalizer because £j. • 6fcJ — bki. Show that this representation is equivalent
to the representation U\.

2. Let H be an infinite-dimensional separable Hilbert space, A « ^(H)  the Banach
algebra of Hilbert-Schmidt operators on H, and B a closed self-adjoint subalgebra ofogically simple.